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<title>Atlas software user guide -- Strong real forms</title>
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<h2>Strong real forms</h2>
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<i>Last updated: October 15, 2005</i>
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Let G be a connected reductive complex algebraic group. A <i>pinning</i> of
G is a triple (T,B,{X<sub>&alpha;</sub>}<sub>&alpha;&#8712;&Pi;</sub>), where
T is a maximal torus in G, B is a Borel containing T, &Pi; is the set of
simple roots of (G,T) with respect to B, and for each &alpha;&#8712;&Pi;,
X<sub>&alpha;</sub> is a root vector for &alpha; in the Lie algebra of G; we
fix such a pinning &#8472; (containing our already chosen T and B) in all that 
follows. It is well-known that the adjoint group Int(G)=G/Z acts simply 
transitively on the set of pinnings. It follows that the exact sequence
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1 &#8594; Int(G) &#8594; Aut(G) &#8594; Out(G) &#8594; 1
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splits by identifying Out(G) with the stabilizer of &#8472; in Aut(G). In
particular, each <a href="innerclass.html">inner class</a> of real forms of G
has a canonical representative &theta;<sub>f</sub> fixing &#8472;; we will
say that &theta;<sub>f</sub> is the fundamental involution in the inner class.
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Let &Gamma;=<b>Z</b>/2, and write &Gamma;={1,&delta;}. Then we may form the
semidirect product
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G<sup>&Gamma;</sup> = G&#215;&#124;{1,&delta;}
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where the conjugation action of &delta; on G is through &theta;<sub>f</sub>.
This is an analogue in our situation of the Langlands group, together with
a specific choice of splitting; we will be mostly concerned with the
non-identity component G.&delta; in G<sup>&Gamma;</sup>.
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A <i>strong involution</i> in G<sup>&Gamma;</sup> is an element x in G.&delta;
such that x<sup>2</sup>&#8712;Z(G); a <i>strong real form</i> of G is a
G-conjugacy class of strong involutions. Each strong involution x defines an
ordinary involution &theta;<sub>x</sub> of G by setting 
&theta;<sub>x</sub> = int(x); it is clear that in this way we obtain exactly
all involutions in the inner class defined by &theta;<sub>f</sub>. By passing
to conjugacy classes, to each strong real form corresponds a real form as
defined <a href="realforms.html">here</a>. When G is adjoint, the notions of
real form (in the chosen inner class) and strong real form coincide.
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Let W<sub>im</sub> be the Weyl group of the imaginary root system for the
fundamental involution &theta;<sub>f</sub>. As mentioned 
<a href="realforms.html">here</a>, the classification of strong
real forms reduces essentially to an orbit computation for an action of
W<sub>im</sub>. Precisely, the strong involutions of G may be partitioned
according to the values of x<sup>2</sup>=z &#8712; Z(G); for a fixed value
of z, there is a simply transitive action of the 
<a href="tori.html">component group</a> of the torus dual to the fundamental
torus of G, on the set of strong involutions with square z that induce on
T the same involution as &theta;<sub>f</sub>. Then the orbits of W<sub>im</sub>
on that set of involutions classify the strong real forms of G with square z.
Moreover, multiplication by elements of Z will obviously induce isomorphisms
among &#8220;packets&#8221; of strong real forms; so packets
that have the same image in the adjoint group give rise to the same orbit
problem, and it is really enough to solve it once for each of the &#8220;packet
images&#8221; in the adjoint group. This is what the &#8220;strongreal&#8221;
command does; the classes of real forms mentioned in its output are the packet
images. Then all that remains to be done to complete the classification is
to count the number of packets, which may be in  fact be infinite for some
reductive groups (precisely, it is infinite if and only if the radical of
G contains a compact factor.)
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By a similar procedure, one may also classify the strong real forms for which
any given conjugacy class of <a href="cartan.html">Cartan subgroups</a> is 
defined. The argument to the &#8220;strongreal&#8221; command specifies the
Cartan class.
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